Ensemble-Level Upscaling for Efficient Estimation of Fine ... - OnePetro

Ensemble-Level Upscaling for Efficient Estimation of Fine-Scale Production Statistics Yuguang Chen, SPE, Chevron Energy Technology Company, and Louis J. Durlofsky, SPE, Stanford University

Summary Upscaling is often needed in reservoir simulation to coarsen highly detailed geological descriptions. Most existing upscaling procedures aim to reproduce fine-scale results for a particular geological model (realization). In this work, we develop and test a new approach, ensemble-level upscaling, for efficiently generating upscaled two-phase flow parameters (e.g., upscaled relative permeabilities) for multiple geological realizations. The ensemble-level upscaling approach aims to achieve agreement between the fineand coarse-scale flow models at the ensemble level, rather than realization-by-realization agreement, as is the intent of existing upscaling techniques. For this purpose, flow-based upscaling calculations are combined with a statistical procedure based on a cluster analysis. This approach allows us to compute numerically the upscaled two-phase flow functions for only a small fraction of the coarse blocks. For the majority of blocks, these functions are estimated statistically on the basis of single-phase velocity information (attributes), determined when the upscaled single-phase parameters are calculated. The procedure is designed to maintain close correspondence between the cumulative distribution functions (CDFs) for the numerically computed and statistically estimated two-phase flow functions. We apply the method to 2D synthetic models of multiple realizations for uncertainty quantification. Models with different geological heterogeneity and fluidmobility ratios are considered. It is shown that the method consistently corrects the biases evident in primitive coarse-scale predictions and can capture the ensemble statistics (e.g., P50, P10, P90) of the fine-scale results almost as accurately as the full flow-based upscaling procedures but with much less computational effort. The overall approach is flexible and can be used with any combination of upscaling procedures. Introduction In recent years, a wide variety of upscaling procedures has been developed and applied. These techniques generally take as their starting point a fine-scale geological model of the subsurface. The intent is then to generate a coarser model, which retains the geological realism of the underlying fine-scale description, for use in flow simulation. Though model sizes can vary substantially depending on the application, typical fine-scale geocellular models may contain 107 to 108 cells, while typical simulation models may contain 104 to 106 blocks. Recent reviews and assessments (e.g., Barker and Thibeau 1997; Barker and Dupouy 1999; Farmer 2002; Darman et al. 2002; Gerritsen and Durlofsky 2005; Chen 2005) describe and apply a variety of upscaling techniques. These procedures can be categorized in different ways. One important distinction is in terms of the coarse-scale parameters that are computed by a particular method. Specifically, a technique that generates only upscaled single-phase parameters (permeability or transmissibility) can be classified as a single-phase upscaling procedure even though it may be applied to two- or three-phase flow problems. A method that additionally

This paper (SPE 106086) was accepted for presentation at the 2007 Reservoir Simulation Symposium, Houston, 26–28 February, and revised for publication. Original manuscript received for review 6 December 2006. Revised manuscript received for review 17 January 2008. Paper peer approved 5 February 2008.

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generates upscaled relative permeability functions is termed a twophase upscaling procedure. Another way to distinguish upscaling procedures is according to the problem solved to determine the coarse-scale parameters. In particular, methods may be classified as local, extended local, quasiglobal, or global in order of increasing computational effort, depending on the problem solved in the upscaling computations. In general, two-phase upscaling methods are more computationally expensive than single-phase upscaling procedures, as a time-dependent two-phase flow problem must be solved in this case. The appropriate upscaling procedure for any particular problem depends on the required level of accuracy and the degree of coarsening. For example, for permeability fields characterized by twopoint geostatistics (variogram-based models), with only a moderate degree of coarsening, the use of local single-phase upscaling procedures, possibly coupled with nonuniform gridding, may provide acceptable coarse models. For more challenging cases, however, such as channelized systems characterized by multipoint geostatistics and high degrees of upscaling, extended local or (quasi) global single-phase upscaling coupled with two-phase upscaling may be necessary. In recent work (Chen and Durlofsky 2006b), we introduced an upscaling procedure that combines quasiglobal single-phase upscaling, which was accomplished through a local-global procedure, with a specialized two-phase upscaling. The technique was shown to provide reasonable degrees of accuracy for challenging problems, though it was observed that the speedups between finegrid simulation and the upscaling plus coarse-scale simulations were not that dramatic (e.g., approximately a factor of 4 to 10). Speedups will be much more substantial if the model is simulated many times, because the computation time required for the twophase upscaling calculations is large compared to the coarse-grid simulations. It would, however, still be useful to accelerate these upscaling computations. This is particularly desirable in cases with substantial uncertainty in the underlying geological model, in which case many realizations (or scenarios) are to be simulated. In such cases, realization-by-realization agreement between fine and coarse models is less essential. Rather, what is required in this case is agreement of a statistical nature, such as agreement in the CDFs (e.g., the P10, P50, P90 predictions) for relevant production quantities such as cumulative oil recovered or net present value. The required level of accuracy of the upscaling, on the realization-byrealization basis, could be slightly less for such cases, though the method should be unbiased. The intent of this paper is to develop and test procedures for substantially accelerating two-phase upscaling procedures for cases in which many realizations are to be considered. Toward this goal, we couple upscaling with statistical estimation techniques. Several statistical techniques were considered, though the best performance was achieved using K-means clustering. Application of this approach allows us to compute upscaled two-phase functions through full-flow simulation for only a small fraction of the coarse-scale blocks. For the rest of the blocks, these functions are estimated statistically on the basis of velocity information (attributes) computed during the single-phase upscaling. The overall method can be used with any combination of single-phase and two-phase upscaling procedures and is shown to provide a high level of accuracy in the statistical sense described above for example cases involving different heterogeneity models. December 2008 SPE Journal

There has been very little research reported on the development of upscaling procedures for multiple permeability realizations. Previous researchers considered related problems involving the handling of upscaled multiphase flow parameters (e.g., the grouping of pseudorelative permeabilities). Dupouy et al. (1998) applied a statistical procedure to group the numerically computed global pseudorelative permeabilities to reduce the number of pseudofunctions used in flow simulation. Their work did not involve the estimation of upscaled relative permeabilities, though they noted that such an approach would be useful in practice because it would reduce the number of pseudofunctions to be numerically computed. Christie and Clifford (1998) suggested an a priori approach to grouping upscaled parameters for compositional simulation. They used the concept of tracer-breakthrough curves to represent coarse-scale blocks, and applied K-means clustering analysis to group the upscaled functions. Neither of these studies, however, considered upscaling over multiple reservoir models and the associated assessment of uncertainty for fine-scale predictions. Our work here is also related to previous studies on error modeling of coarse-scale simulation models (Omre and Lødøen 2004, Lødøen et al. 2004), though the approaches are quite different. In the error modeling studies, some fine-scale calibration runs were required to model upscaling error and correct the bias in the coarse-scale simulation results, while our approach here estimates the upscaled flow parameters directly. The statistical estimation procedure (based on cluster analysis) used here can be viewed as a proxy or surrogate method that avoids the need to numerically generate upscaled two-phase parameters. In this sense, any proxy can be applied in the procedure. Statistical clustering approaches are used in many applications and have been applied recently in reservoir engineering as proxies for simulations in genetic algorithm-based optimization (Artus et al. 2006). The outline of this paper is as follows: We first provide the governing equations and a brief overview of the relevant upscaling procedures. Next, we describe and illustrate the ensemble-level upscaling approach based on clustering to estimate statistically the upscaled two-phase flow functions. This is followed by extensive numerical results for a variety of 2D systems. We conclude with a discussion and summary. Overview of Upscaling Methodology Governing Fine-Scale Equations. We consider viscousdominated oil and water two-phase systems and, for convenience, neglect the effects of capillarity, gravity, and compressibility. The governing equations are formed by combining mass conservation equations with Darcy’s law for both phases, and they can be expressed in terms of the usual pressure and saturation equations: ⵜ ⭈ 关␭共S兲k共x兲 ⭈ ⵜp兴 = 0 and

␾

Coarse-Scale Simulation Models. As discussed in the Introduction, coarse models for two-phase flow can be constructed using only upscaled single-phase parameters or using both upscaled single-phase and two-phase parameters. Here, our emphasis is on highly coarsened models. Specifically, we upscale fine-scale models that are of dimensions 100×100 to 10×10 coarse models, so we will need to compute both types of upscaled parameters. Upscaled single-phase flow quantities include equivalent permeability k* or transmissibility T*, where the superscript * denotes upscaled quantity, while coarse-scale two-phase parameters entail upscaled relative permeability functions krw*(Sc) and kro*(Sc), or analogously ␭*(Sc) and f *(Sc), where Sc designates the coarse-scale saturation. In this work, the coarse-scale flow model is taken to be the same as the fine-scale model (Eq. 1), with the coarse parameters k*, ␾*, ␭*, and f* replacing the fine-scale functions: ⵜ ⭈ 关␭*共Sc兲k*共xc兲 ⭈ ⵜpc兴 = 0 and

␾*

⭸Sc + ⵜ ⭈ 关ucf*共Sc兲兴 = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2) ⭸t

Here, the superscript c indicates coarse-scale variable. The quantities computed are now the coarse-scale pressure pc and saturation Sc. In this work, porosity is taken as constant. The primitive coarse-scale model, considered below, involves only upscaled single-phase parameters. Specifically, we compute k* or T* (and ␾*) but retain the fine-scale relative permeability functions, i.e., f*⳱f and ␭*⳱␭ in Eq. 2. More general upscaled models for two-phase flow, which include coarse-scale dispersive terms, are also available and provide improved accuracy in some cases (e.g., Efendiev and Durlofsky 2004, Chen and Durlofsky 2006b) but will not be considered here. Flow-based calculation of the upscaled quantities entails appropriate fine-scale simulations. The upscaling procedures applied in this work are described in detail in our earlier work (Chen et al. 2003; Chen and Durlofsky 2006a and 2006b; Wen et al. 2006), so the descriptions here are brief. In general, the accuracy of any upscaling procedure (for single-phase or two-phase flow parameters) depends on the problem that is solved (domain and boundary conditions) to determine the coarse-scale parameters. Following this solution, appropriate averages are computed and the upscaled parameter, which relates these averaged quantities, is determined. The simplest and most computationally efficient approaches use local regions and apply generic boundary conditions. By “local region,” we mean the portion of the fine-scale model corresponding to the target coarse-scale gridblock, or the two coarse blocks sharing the target coarse-block interface in the case of transmissibility upscaling. A schematic showing the global fine- and coarsescale grids and a local region for upscaling is displayed in Fig. 1. Better accuracy can often be achieved by using larger computational domains for the upscaling. Examples of this are extended

⭸S + ⵜ ⭈ 关uf共S兲兴 = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (1) ⭸t

Here, source terms are omitted for simplicity. In Eq. 1, k is the (diagonal, spatially variable) absolute permeability tensor, p is pressure, S is water saturation, ␾ is porosity, x represents physical space, and t is time. Additional quantities are ␭(S), the total mobility, defined as the sum of water and oil mobilities, i.e., ␭⳱␭w+␭o⳱krw/␮w+kro/␮o, where krw(S) and ␮w are the relative permeability to water and water viscosity, and kro(S) and ␮o are analogous quantities for oil. The Buckley-Leverett fractional flow function f(S) is defined as f⳱␭w/(␭w+␭o), and the total Darcy velocity u is computed by u⳱−␭k⭈ⵜp. In this work, the system is solved using an implicit pressure, explicit saturation (IMPES) procedure. The pressure equation is solved using a standard finite difference, or, more properly, finite volume method, while the saturation equation is solved using a second order total variation diminishing (TVD) technique. See Chen (2005) for full details on these methods. December 2008 SPE Journal

Fig. 1—Schematic showing global fine-scale and coarse-scale grids and local region, containing target coarse-block interface, for upscaling calculations. 401

local procedures, in which the local domain is extended to include additional portions of the fine-scale model, and global and quasiglobal procedures, in which the full global domain is used. In global methods, the full fine-scale model is used for the upscaling computations, while in quasiglobal approaches an approximate global solution is applied instead. Upscaled Single-Phase Flow Parameters. Accurate single-phase upscaling is a prerequisite for coarse-scale two-phase or multiphase flow models. We recently developed a family of procedures, called local-global methods, for the determination of upscaled single-phase parameters. These approaches use global coarse-scale information to supply boundary conditions for extended local upscaling computations. The procedure is iterated until a self-consistent solution is achieved. In our earlier work with local-global procedures (Chen et al. 2003), we observed better accuracy in coarse-scale models by computing upscaled transmissibility T* directly, rather than first computing k* and then calculating T* from k*. Similar findings were reported by other researchers (Romeu and Noetinger 1995). Consistent with those findings, in this work we apply transmissibility upscaling procedures. In this work, we apply both local-global and global transmissibility upscaling techniques to ensure the accuracy in the upscaled single-phase parameters. We note that the use of global singlephase upscaling techniques is viable in practice if the fine model is of a manageable size (i.e., if the steady-state incompressible single-phase pressure equation ⵜ⭈[k(x)⭈ⵜp]⳱0 can be solved with modest computational effort). If this solution is not practical, then local-global or (extended) local transmissibility upscaling procedures must be used. Global procedures have been developed and applied by a number of researchers (e.g., Holden and Nielsen 2000, Zhang et al. 2005). Upscaled Two-Phase Flow Functions. The determination of upscaled two-phase functions (␭* and f* , or equivalently krj*, where j denotes the phase) is often more computationally demanding than the calculation of T*. If global two-phase upscaling methods are applied, time-dependent fine-scale solutions of Eq. 1 are required (i.e., we must simulate the full fine-scale model). However, such methods may be viable for use in conjunction with our statistical estimation approach if many realizations are to be considered and if the simulation of Eq. 1 on the fine scale for a few of these realizations is possible. Global two-phase upscaling methods have been used (e.g., Dupouy et al. 1998, Darman et al. 2002) and have been implemented in commercial software (Schlumberger 2004). More efficient local and extended local two-phase upscaling procedures are often the preferred methods for large problems. For these techniques, it has been shown that the boundary conditions imposed on the (extended) local problems can affect the accuracy of the resulting coarse-scale model. Consider the local domain shown in Fig. 1b. This domain includes the fine-scale cells associated with coarse blocks i+and i–. For the determination of ␭* and f* (or krj*) at coarse-block interface i, pressure and saturation boundary conditions are required at the left edge and pressure boundary conditions are required at the right edge; no flow conditions are prescribed at the top and bottom. Standard boundary conditions, which are often used for this computation, are as follows: p共0, y兲 = 1, p共Lx, y兲 = 0, S共0, y兲 = 1 and uy共x, 0兲 = uy共x, Ly兲 = 0, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3) where Lx and Ly designate the size in the x and y dimensions of the region under consideration. In incompletely layered systems, it has been shown that these boundary conditions can act to overestimate the movement of injected fluid and, as a result, may give coarse models that predict the early breakthrough of injected fluids and overestimate the total flow rate (Wallstrom et al. 2002, Hui et al. 2005, Chen and Durlofsky 2006b). Specifically, for systems that are layered at the scale of the coarse block, these boundary conditions give ux∼kx, where kx is the fine-scale permeability at boundaries. In an attempt 402

to compensate for this effect, new boundary conditions, called effective flux boundary conditions (EFBCs), were devised by Wallstrom et al. (2002). These boundary conditions view the inlet and outlet blocks as though they were inclusions immersed in a background of equivalent large-scale permeability kb. Here kb can be thought of as the equivalent global permeability for the entire system. EFBCs act to attenuate the flow rates associated with the high-permeability features. The specific boundary conditions for the system shown in Fig. 1b are ux共0, y兲 =

kx共0, y兲 , kx共0, y兲 + kbx

ux共Lx, y兲 =

kx共Lx, y兲 , kx共Lx, y兲 + kbx

and uy共x, 0兲 = uy共x, Ly兲 = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4) The specific form of Eq. 4 corresponds to the case of circular inclusions, as discussed in Wallstrom et al. (2002). If the inclusion is considered to be a general ellipse, additional EFBC parameters appear in Eq. 4. In addition, in the case of a system that is layered at the global scale, the appropriate boundary conditions are essentially those given by the standard boundary specification of Eq. 3. Other boundary specifications for saturation are also possible and may lead to improved results in some cases. For complex permeability fields, it is not always clear a priori which particular boundary specification is optimal for the twophase upscaling problem. Within the context of ensemble-level upscaling, appropriate boundary conditions can be determined by comparing fine- and coarse-scale simulation results for one or more realizations. If it is impractical to simulate even a single fine-scale model, a portion of the model can be considered. From the discussion in this section, it is evident that there are a variety of techniques for both single-phase parameter and twophase parameter upscaling. As discussed in Chen and Durlofsky (2006b), it is possible to use essentially any combination of singleand two-phase upscaling techniques (e.g., local-global T* coupled with extended local ␭*(Sc) and f*(Sc) computed using EFBCs). We note that different combinations will be optimal for different problems and computational budgets. The focus of this work is to develop a procedure for the fast estimation of two-phase flow parameters, which is suited for capturing relevant ensemble statistics of multiple fine-scale simulations. For this purpose, we view the determination of the appropriate upscaling method for the problem at hand as essentially a key preprocessing step. Depending on what is practical computationally, extended local, local-global, or global methods can be used to compute T*. Similarly, if it is feasible to perform one or more fine-scale two-phase flow simulations, then either global two-phase upscaling can be used or the optimal local two-phase upscaling technique can be determined. Once the particular upscaling methods are established, the statistical estimation techniques described in the next section for the rapid determination of ␭*(Sc) and f*(Sc) can be applied. These procedures are applicable for any of the two-phase upscaling procedures described above. An Ensemble-Level Upscaling Approach We now present a statistical estimation procedure to efficiently generate the upscaled two-phase flow parameters for multiple geological realizations. We refer to this procedure as an ensemblelevel upscaling approach because it aims to capture the ensemble statistics of the fine-scale flow simulation models rather than agreement between the fine- and coarse-scale models on a realization-by-realization basis. In this procedure, full flow-based upscaling calculations are performed only for a fraction of coarse blocks in the model (e.g., several realizations). These numerically calculated two-phase parameters (␭* and f*) are then calibrated to some readily computed (or already available) coarse-block attributes. This allows ␭* and f* in other coarse blocks to be estimated statistically, thus avoiding flow-based upscaling calculations. December 2008 SPE Journal

Fig. 3—Two realizations of variogram-based permeability field (lx=0.4, ly=0.01, ␴=2.0). Log scale for permeability.

Fig. 2—Fine-scale relative permeabilities and two-phase flow functions for M=5.

Parameterization of Upscaled Two-Phase Flow Functions. We first present a parameterization to represent the two-phase flow functions ␭*(Sc) and f*(Sc). The parameterization of pseudorelative permeabilities has been studied by a number of researchers. For example, Saad et al. (1995) suggested the use of endpoint relative permeability values. Christie (1996) proposed three parameters to characterize the fractional flow and total mobility curves, namely shock-front saturation, slope of the fractional flow curve, and the minimum value of the total mobility. Dupouy et al. (1998) considered several approaches, including the parameters suggested by Christie (1996), discretized pseudorelative permeability curves, and a principal component analysis of the discretized curves. Dupouy et al. (1998) also suggested the use of functional models but noted that because of the wide variety of pseudorelative permeability shapes, this may require very flexible functional forms. Our approach in this work is quite different from those in earlier studies. Previous researchers focused on the direct representation of the upscaled krj* (or ␭* and f*) curves, while our approach is to quantify the difference between the input fine-scale functions (rock curves) and the upscaled functions. This is motivated by the fact that, for a particular set of fluid properties, the deviation of the upscaled function from the rock curve is mainly driven by the underlying fine-scale heterogeneity. We consider two-phase flow with krw⳱S2 and kro⳱(1−S)2, as displayed in Fig. 2a. For simplicity, we take the connate water (Swc) and residual oil (Sor) saturations to be zero, and this set of relative permeabilities is applied to the entire reservoir model. The ensemble-level upscaling approach is not constrained to these assumptions and can be applied to more general cases. The total mobility ␭ and fractional flow function f are determined by krj and the endpoint water/oil mobility ratio M, where M⳱␮o/␮w, because the endpoint relative permeability values are 1. For a typical oil/ water flow (M⳱5), the ␭ and f functions are shown in Figs. 2b and 2c, respectively. In this study, ␭*(Sc) and f*(Sc) are computed directly, because these are the terms that appear in Eq. 2. Thus, we do not actually construct the krj*(Sc), though this can be immediately accomplished if necessary. Note that ␭* and f* are treated as directional (x and y) quantities. For a partially layered system, as shown in Fig. 3 (left), the upscaled ␭* and f* in the x direction for two December 2008 SPE Journal

coarse-block interfaces are displayed in Fig. 4. These are obtained using local two-phase upscaling calculations with EFBCs. In this work, for all of the locally computed ␭* and f*, we use a domain consisting of two coarse blocks and compute ␭* and f* at the coarse-block interface, as depicted in Fig. 1b. Note that, although these functions are associated with interfaces, we still often refer to them as “coarse-block” functions. We see that for a given Sc, the upscaled functions give higher values of the total mobility and fractional flow function. As is typical for upscaled two-phase functions, these act to correct the late breakthrough often encountered in primitive coarse-scale flow models. We quantify the difference between the input fine-scale and upscaled functions by discretizing both the fine-scale and upscaled functions. As illustrated in Fig. 5, these functions are discretized at N (here N⳱20) equally spaced saturation values between 0 and 1 or, more generally, between Swc and 1−Sor. Then the difference between ␭* and ␭ (or between f* and f ) can be quantified as

␦␭x =

1 N

N

兺 | ␭*共S 兲 − ␭共S 兲 | c i

x

c i

i=1

and 1 ␦fx = N

N

兺 | f *共S 兲 − f共S 兲 |, x

c i

c i

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5)

i=1

Fig. 4—Upscaled total mobility ␭* and fractional flow function f * (in the x direction) for two coarse-block interfaces for variogram-based model (lx=0.4, ly=0.01, and ␴=2.0), as shown in Fig. 3. 403

In a recent study in which the ensemble-level upscaling was extended and applied to 3D well-driven flows (Chen et al. 2008), we investigated in more detail the correlation between coarseblock attributes and the upscaled two-phase functions. Additional attributes considered there included the coefficient of variation (defined by ␴/〈u〉) and the integral ranges of the autocorrelation functions of the velocity and permeability fields. It was found that the use of attributes somewhat different from those used in this study (described in the next subsection) improved the clustering for 3D well-driven flows. In this work, we use 〈u〉 and ␴u as coarse-block attributes.

Fig. 5—Parameterization of upscaled two-phase flow functions.

where Sci designates a particular saturation value. The quantities in the y direction (␦␭y and ␦fy) are defined analogously. This parameterization uses only one parameter to represent each upscaled function or, more precisely, the difference between the upscaled and fine-scale functions. For the case shown in Fig. 5, ␦␭x⳱0.184 and ␦fx⳱0.175, while for those shown in Fig. 4, ␦␭x⳱0.189 and ␦fx⳱0.211 for Fig. 4a and ␦␭x⳱0.046 and ␦fx⳱0.087 for Fig. 4b. As pointed out previously, these values vary on the basis of the fine-scale permeability heterogeneity. We note that this parameterization does not precisely distinguish individual functions, as two different upscaled curves may give the same (or very similar values) of ␦␭ (or ␦f). However, it should be kept in mind that in our ensemble-level upscaling approach, the upscaled two-phase functions are estimated statistically with the ultimate goal of capturing the ensemble statistics of the fine-scale models. Thus, an exact representation may not be necessary here. In fact, these one-parameter representations appear to be sufficient for current purposes because they allow us to readily quantify and preserve appropriate statistics (e.g., the CDFs) for the upscaled two-phase functions. Coarse-Scale Block Attributes. When the parameterization is applied to numerically simulated ␭* and f*, the resultant parameters ␦␭ and ␦f can be quite different for different coarse-block interfaces, depending on the different permeability distributions within coarse-scale blocks. Our objective now is to establish some type of correlation between the upscaled two-phase flow functions and the coarse-scale blocks. Toward this end, we define appropriate coarse-block attributes that are indicative of the effect of the underlying permeability heterogeneity on flow but are at the same time already available or very inexpensive to compute. The attributes used here are the average 〈ux〉 and standard deviation ␴ux of the single-phase fine-scale velocity, computed over the region corresponding to the two coarse blocks that share the interface at which ␭x* and fx* are required. Note that we refer here to an x-oriented interface, though the procedure is analogous for a y-oriented interface. These attributes are a natural choice for two key reasons. First, the fine-scale information needed to determine 〈ux〉 and ␴ux is already available from the determination of the upscaled single-phase flow parameters (e.g., Tx*), so there is essentially no cost associated with their computation. Second, it is known from volume averaging of the fine-scale equations (e.g., Efendiev and Durlofsky 2004) that the subgrid effects that appear in the coarse-scale equations are largely driven by fine-scale velocity fluctuations. Because ␴ux is the basic parameter quantifying these velocity fluctuations, it is reasonable to expect that it will be an appropriate attribute. The average velocity is also a useful attribute because it provides a direct indication of the impact of permeability on flow. In this study, we have also considered other attributes, such as the largest several components of a principal component analysis or kernel principal component analysis of the fine-scale velocity or permeability fields within coarse-scale blocks. Limited investigations showed that 〈ux〉 and ␴ux outperformed those quantities, and those based directly on permeability. 404

Statistical Estimation of Upscaled Two-Phase Flow Functions. With the parameterization of upscaled two-phase functions and the definition of coarse-block attributes, we are now in a position to define a procedure to assign upscaled two-phase flow functions to coarse-scale blocks. As indicated in the Introduction, a variety of proxy models exist that may, in principle, be applied in the ensemble-level upscaling. In this work, we employ a cluster-analysis technique that belongs to the general category of classification methods. Regression techniques, which take coarse-block attributes as input and provide a continuous output of the upscaled two-phase flow functions (␦␭ or ␦f), may also be applied. In the recent study of 3D well-driven flows (Chen et al. 2008), we introduced geostatistical simulation methods (e.g., sequential Gaussian simulation) in the ensemble-level upscaling, which account for spatial correlation in the upscaled functions. In this work, we will describe only the basic procedure. In this paper, we use the term “statistical estimation” to describe our procedure. We note, however, that in the context of stochastic simulation and modeling (Goovaerts 1997), our approach can be viewed as a “conditional simulation” technique, because it reproduces the CDF of the upscaled two-phase flow functions (␦␭ and ␦f). However, because we use the term “numerically simulated” in the context of flow simulation, we refer to our method as a “statistical estimation” procedure to avoid confusion. Calibration Based on Cluster Analysis. We will use the heterogeneous permeability fields of the type shown in Fig. 3 to illustrate the statistical procedure for the calibration and estimation of the upscaled two-phase functions. We consider 100 fine-scale realizations of a partially layered permeability field. Detailed descriptions of this model will be presented in the Numerical Results section. We numerically compute ␭* and f* in both the x and y directions for 10 realizations of the fine-scale model using flowbased two-phase upscaling. For this example, global two-phase upscaling is applied. Our objective is to statistically estimate ␭* and f* for the other 90 realizations such that the resultant ␭* and f* functions have unbiased CDFs with reference to those computed from numerical simulations. The parameterization described above is applied to all the numerically simulated ␭* and f*, and the coarse-block attributes (〈u〉 and ␴u) are computed for all the coarse-blocks for the 10 realizations. We apply an unsupervised cluster analysis to the coarseblock attributes and compute the corresponding CDFs for the upscaled two-phase flow functions, represented here by ␦␭ and ␦f. A K-means clustering technique is applied to divide the data in the attribute space into different classes. Shown in Fig. 6 is the partitioning of the attribute data (〈ux〉 and ␴ux) into three classes, indicated by different shapes and shades. In the K-means clustering method, the number K of centers is predefined and the distance of each data point from the center of its cluster is minimized iteratively. In Fig. 6, the encircled black crosses designate the cluster centers in the final partition. For more details on the clustering technique and its application, see Bishop (1995) and Artus et al. (2006). In Fig. 6, there are 1,000 data points (10 realizations, each with 100 coarse-block interfaces). Cluster 1 designates coarse blocks with higher 〈ux〉 and ␴ux, while Cluster 3 corresponds to lower values. Note that the clustering is based on the coarse-block attributes and does not use the parameters for the upscaled twophase functions (␦␭x and ␦fx). However, an experimental CDF of December 2008 SPE Journal

Fig. 6—Partitioning of data points (from 10 realizations) into three clusters in the attribute space. The data are from variogram-based model, as shown in Fig. 3.

␦␭x and ␦fx can be readily computed for each cluster. The resultant CDFs are presented in Fig. 7, where there is a wide range of values of ␦␭x and ␦fx, indicating a variety of upscaled ␭x* and fx* functions. Of interest is the statistical correspondence between the distribution of ␦␭x and ␦fx and the attributes (i.e., higher velocity variations correspond to larger values of ␦␭x and ␦fx, as would be expected). This cluster analysis also was applied to channelized models, for which the flow results will be presented in the next section. For the channelized case, we observed clustering results similar to those shown in Fig. 6. We note that the two attributes shown in Fig. 6 (〈ux〉 and ␴ux) are somewhat correlated. As indicated above, these clustering results can likely be improved through the use of other or additional attributes. This was discussed in a recent paper (Chen et al. 2008). In this work, we also considered the use of the average and standard deviation of the fine-scale permeabilities as the coarse-block attributes, but these did not distinguish the CDFs among different clusters as clearly as the velocity attributes. Here, the CDFs are referred to as the calibration CDFs and will be used to statistically estimate the ␦␭x and ␦fx for coarse-block interfaces in other realizations. Statistical Estimation Based on Calibration CDFs. Given a coarse-block interface for which ␭x* and fx* are to be estimated, we can determine the target cluster based on the coarse-block attributes 〈ux〉 and ␴ux. Specifically, if the new data point is closest to the center of Cluster 2 in the attribute space, then this coarseblock interface is assigned to Cluster 2. Accordingly, the calibration CDFs in Cluster 2 (dashed curves in Figs. 7a and b) are used to assign ␦␭x and ␦fx to this coarse-block interface. This is performed by applying the inverse CDFs to a random number generated from a uniform distribution. Given enough data points, this ensures that the estimated ␦␭x and ␦fx for a particular cluster honor the calibration CDFs for that cluster. We verified that the CDFs of the estimated parameters ␦␭x and ␦fx reproduce the true CDFs computed from the numerically simulated ␦␭x and ␦fx. Note that a uniform random function is applied here. In our recent work, the basic procedure was extended to use geostatistical methods in the generation of ␦␭x and ␦fx. For example, sequential Gaussian simulation can be introduced by using a Gaussian random function and taking into account the spatial correlation in the upscaled functions. Other variants are also possible (e.g., the joint estimation of ␦␭x and ␦fx). The flow results in the next section demonstrate that the basic procedure as applied here does provide accurate predictions of the ensemble statistics of fine-scale flow results. More sophisticated approaches, such as those referred to previously, may provide more accurate results in some cases. The final step is to transfer the statistically estimated ␦␭x and ␦fx back to functional forms to be used in flow simulations. For a particular ␦␭x (or ␦fx), we simply search the target cluster for the closest values of the calibration ␦␭x (or ␦fx). This functional form is then assigned to the target coarse-block interface. Note that, December 2008 SPE Journal

Fig. 7—Calibration CDFs, determined from data shown in Fig. 6.

although the assignment of the functional form is deterministic, this step uses the stochastic estimation of the parameters ␦␭x and ␦fx. Thus, the overall procedure is a statistical estimation. A similar procedure is applied to estimate ␦␭y and ␦fy, with attributes being the velocity information in the y direction. For this example, the values of ␦␭y and ␦fy are much smaller than those in the x direction because of the layered structure of the permeability field. From the above description, it is clear that in the ensemblelevel upscaling approach, there is not a one-to-one correlation but rather a statistical correlation between the coarse-block attributes and the upscaled two-phase flow functions. Becasue our objective here is an ensemble-level agreement between the fine- and coarsescale models, the procedure is designed to honor the CDFs of the upscaled two-phase functions. Although this does provide an unbiased estimation of these functions, the resulting ␭* and f* cannot be expected to be highly accurate for each individual coarse-block interface. Nonetheless, as will be shown in the next section, this procedure is effective for efficiently providing upscaled two-phase flow functions for multiple geological realizations and for capturing the ensemble-level behavior of the fine-scale models. Numerical Results We now present results for three example cases. These 2D systems include two variogram-based models with different correlation structures and a channelized model. For all cases, we consider 100 realizations of the fine-scale model, and the goal is to reproduce the ensemble statistics (e.g., P50 prediction and P10–P90 interval) using the coarsened models. Several different upscaling procedures are applied. Variogram-Based Model with lx=0.4, ly=0.01, and ␴=2. The first case involves a log normally distributed permeability field. 405

Fig. 8—Flow results for 100 fine-scale models for variogrambased model (lx=0.4, ly=0.01, and ␴=2.0) and M=5. Curves represent P50 (solid curve) and P10–P90 interval (dashed curves).

This case was used earlier to illustrate the ensemble-level upscaling procedure, and two realizations were shown in Fig. 3. The model domain is a square of side length L, and it is characterized by dimensionless correlation lengths lx⳱0.4 and ly⳱0.01, defined as the dimensional correlation length divided by L. The model possesses a high degree of variability (␴⳱2, where ␴2 is the variance of log k). The permeability distributions are not conditioned to any data and are generated using sequential Gaussian simulation (Deutsch and Journel 1998) with a spherical variogram model. For these runs, we set M⳱␮o/␮w⳱5. The simulations in this and subsequent examples are performed by fixing the pressure at the left and right edges of the model to different constant values. We first perform fine-scale simulations for the 100 realizations. These results are shown in Fig. 8, where we plot total (oil plus water) flow rate (Qx) at the inlet or outlet edge (these rates are the same because the system is incompressible) vs. time and oil cut (Fo) vs. time for all realizations. Time, here and in subsequent figures, is nondimensionalized by a characteristic time that is the same for all realizations. A reasonable degree of variation is evident. In Figs. 8a and b, the solid black curves designate P50 flow responses while the dashed black curves designate the P10 (lower curves) and P90 (upper curves) predictions. The P10 and P90 curves are computed such that 10% of the data falls below the P10 curves and 10% of the data falls above the P90 curves. Thus, 80% of the data lies between these two curves. Next the P50, P10, and P90 flow predictions of the fine- and various coarse-scale models will be compared. We note that if the instantaneous oil and water flow rates are well preserved, integrated quantities such as cumulative oil production will also be reproduced. In the first set of results, we apply global single- and two-phase upscaling techniques. This is done in order to minimize the upscaling error so we can focus on the performance of the statistical estimation procedure. In later examples, we apply less expensive, and accordingly less accurate, quasiglobal and extended local upscaling procedures. Results for Qx for the upscaled models are shown in Fig. 9. In this and all subsequent figures, the solid curves correspond to the fine-scale results, the dot-dash curves to the results for coarse-scale models, the thick curves designate P50, and the thin curves designate the P10 (lower curves) and P90 (upper curves) predictions. In the primitive coarse-model results (Fig. 9a), we use global T* along with the fine-scale relative permeability curves. For the global pseudofunction results (Fig. 9b), we apply global T* and compute global ␭* and f* numerically for each block in each realization, which requires a full fine-scale simulation for each realization. It is apparent that the primitive coarse-model results show some error, particularly at times less than approximately 0.05, while the global pseudofunction model provides accurate results at all times. Results using the statistical estimation procedure are shown in Figs. 9c and 9d. Here, global upscaling is applied to 10 of the 100 realizations, and the attributes and upscaled parameters from these 10 simulations (corresponding to a total of 1,000 coarse-block interfaces) are used to populate the clusters, as described previously. Using these clusters for the statistical estimation of ␭* and f* , we generate the results shown in Figs. 9c (using three clusters) 406

Fig. 9—Comparison of P50 (thick curves) and P10–P90 interval (thin curves) for total flow rate between fine-scale (solid curves) and coarse-scale (dot-dash curves) models for variogrambased model (lx=0.4, ly=0.01, and ␴=2.0) and M=5. For estimated pseudofunctions, 10 realizations are used for calibration.

and 9d (using one cluster). Here, by one cluster, we mean that only one CDF is used to populate ␭* (or f*) for the coarse blocks to be estimated. Both sets of results show reasonably close agreement between the fine- and coarse-scale models for both the P50 and P10–P90 interval. As expected, the results are not quite as accurate as the global pseudofunction results in Fig. 9b, for which all ␭* and f* are computed numerically, but they are clearly superior to the primitive model results in Fig. 9a. Interestingly, there is not much difference between the results for Qx using three clusters and those using one cluster. Fig. 10 displays results for oil cut for the various procedures. Here we see significant errors for the primitive model (Fig. 10a) and again a high degree of accuracy for the global pseudofunction model, for which all blocks in all realizations are upscaled numerically. The results using statistical estimation of the ␭* and f* using three clusters are nearly as accurate as the results for the global pseudofunction model, while some degradation is observed when only one cluster is used. This demonstrates that the use of unbiased ␭* and f* , even when generated from the CDFs of a single cluster, is able to essentially correct the evident bias in the primitive coarse-scale model. These observations are further confirmed by additional results for this case, presented in Figs. 11 and 12. In Fig. 11, we display the CDF of time for a prescribed value of oil cut. These results are generated by drawing a horizontal line at a given value of Fo (here, Fo⳱0.8), intersecting the results for all realizations, which can be visualized with reference to Fig. 8b, and plotting the resulting CDF for time. It is evident from Fig. 11 that the primitive model leads to significant error, as is also apparent in Fig. 10a, and that the global pseudofunction and estimated pseudofunction models provide high degrees of accuracy, though some error is apparent when only one cluster is used. Realization-by-realization comparisons are shown in Fig. 12. Here, we plot the time at which Fo⳱0.8 is reached for each model against the result for the corresponding fine model. A perfect upscaling would result in all points falling on the 45° line. The primitive model leads to substantial error and bias, while the global pseudofunction results illustrate a high degree of accuracy on a realization-by-realization basis. The statistical estimation procedure does not provide as high a degree of accuracy as does the December 2008 SPE Journal

Fig. 11—Comparison of CDF of time between fine-scale and coarse-scale models at Fo=0.8. Variogram-based model (lx=0.4, ly=0.01, and ␴=2.0) and M=5.

Fig. 10—Comparison of P50 (thick curves) and P10–P90 interval (thin curves) for oil cut between fine-scale (solid curves) and coarse-scale (dot-dash curves) models for variogram-based model (lx=0.4, ly=0.01, and ␴=2.0) and M=5. For estimated pseudofunctions, 10 realizations are used for calibration.

global pseudofunction method; but, there is clearly a reasonable correlation, and the predictions appear to be essentially unbiased. We see that, although there is not perfect agreement at the level of individual realizations, the estimation results do reproduce the ensemble statistics of the fine-scale models, as is apparent in the CDF results in Fig. 11, which is exactly the objective of the ensemblelevel upscaling. We note that the results shown in Figs. 11 and 12 are typical of other values of oil cut. The results for this case clearly demonstrate the efficacy of the proposed method. Specifically, these results show that we are able essentially to reproduce the P50 flow behavior and the P10–P90 interval of the fine-scale models, even though we numerically compute the upscaled ␭* and f* for only 10% of the models. Variogram-Based Model With lx=0.5, ly=0.1, and ␴=2. The next case again involves a log normally distributed unconditioned permeability field with ␴⳱2. The correlation structure in this case differs from that in the first example. Two realizations are shown in Fig. 13. In our previous work (Chen and Durlofsky 2006b), it was shown that, for models with this type of correlation structure, the use of EFBCs for the computation of ␭* and f* provided reasonably accurate coarse-scale models. This finding is reiterated

Fig. 12—Realization-by-realization comparison of time at which Fo=0.8 for fine-scale and coarse-scale models. Variogrambased model (lx=0.4, ly=0.01, and ␴=2.0) and M=5. December 2008 SPE Journal

here as we compare the use of EFBCs and standard boundary conditions for the numerical computation of ␭* and f*. Again, we apply the statistical estimation procedure to minimize the number of coarse blocks for which we numerically generate the ␭* and f*. For all the models, adaptive local-global upscaling is used to compute T*. For these runs, we consider a much higher mobility ratio of M⳱50. Fine- and coarse-scale results for Qx are shown in Fig. 14. The primitive coarse model (Fig. 14a) leads to biased results (i.e., underprediction of Qx), while the use of standard boundary conditions for the two-phase upscaling (Fig. 14b) leads to systematic overprediction of Qx, even when all pseudofunctions are computed. These behaviors were observed in our earlier work. The results using statistical estimation based on the standard boundary conditions (Fig. 14c) are quite similar to those in Fig. 14b. This illustrates that, as expected, the statistical estimation procedure in general does not improve upon the results or eliminate the bias from the underlying method—it simply provides a means of generating those results with much less computation. Better accuracy is observed when EFBCs are applied. Using EFBCs for the upscaling, either through numerical computation for all coarse blocks (Fig. 14d) or through the statistical estimation procedure (Fig. 14e), yields results for P50 and for the P10–P90 interval that are in reasonably close agreement with the fine-scale realizations. For the results using statistically estimated ␭* and f *, we again numerically computed the ␭* and f* for all coarse-block interfaces in 10 realizations, and three clusters were used. Consistent with the results for the previous example, very little degradation in accuracy is observed when the statistical estimation procedure is applied. Results for oil cut for this case are comparable to the flow rate results shown in Fig. 14. As observed in previous studies, biased predictions are obtained from both the primitive model (biased toward late breakthrough) and from the coarse model using standard boundary conditions for ␭* and f* (biased toward early breakthrough), while EFBCs provide accurate results. We present the oil cut results in Fig. 15 by comparing the predicted time for a given oil cut (here, Fo⳱0.3) for the coarse-

Fig. 13—Two realizations of variogram-based permeability field (lx=0.5, ly=0.1, ␴=2.0). Log scale for permeability. 407

Fig. 14—Comparison of P50 (thick curves) and P10–P90 interval (thin curves) for total flow rate between fine-scale (solid curves) and coarse-scale (dot-dash curves) models for variogrambased model (lx=0.5, ly=0.1, ␴=2.0) and M=50. For estimated ␭* and f * , 10 realizations and three clusters are used for calibration.

scale and fine-scale models. In Fig. 15, by the designation “local pseudo” either with standard boundary conditions or EFBCs, we mean that ␭* and f* are numerically computed for all coarse blocks in all realizations. Fig. 15a presents the CDF for time at which Fo⳱0.3 is reached. We see that the use of estimated ␭* and f* essentially reproduces the CDF of the corresponding coarse model with numerically simulated ␭* and f*. It is also evident that the results from local EFBCs are in close agreement with the finescale solutions. The corresponding realization-by-realization comparisons are shown in Figs. 15b and 15c. This again demonstrates the biases in the primitive model and coarse model with standard boundary conditions (Fig. 15b) and the unbiased predictions in the coarse model using EFBCs with both simulated and estimated ␭* and f* (Fig. 15c). Channelized Model Conditioned to Seismic Data. Our final example is for a different and more challenging style of geological heterogeneity. Here, we consider a channelized model generated using multipoint geostatistics (Strebelle 2002) for facies (sand and shale) distribution conditioned to seismic data. Unconditioned sequential Gaussian simulation was employed for the permeability distribution within each facies. One realization of the model and the corresponding histogram for permeability are shown in Fig. 16, where the permeability contrast between sand and shale is clear. The coarse models are generated in this case using adaptive localglobal (ALG) upscaling to compute T* and EFBCs for the local computation of ␭* and f*. The mobility ratio for these runs is M⳱5. Fig. 17 displays the oil cuts for the 100 fine-scale realizations and for the primitive coarse-scale models, which use the fine-scale ␭ and f. The solid and dashed black curves again indicate the P50, 408

Fig. 15—CDF and realization-by-realization comparison of time at which Fo=0.3 for variogram-based model (lx=0.5, ly=0.1, ␴=2.0) and M=50.

P10, and P90 results. There is clearly significant variation between the realizations, noticeably more than that for the variogram-based model shown in Fig. 8b, and it appears that the spread is even greater in the primitive-model results. This is confirmed in Fig. 18a, where we plot the P50 and P10–P90 interval for Fo for the fine- and primitive coarse-scale models. In this figure, the bias in the primitive model predictions is clearly evident. Note that our emphasis here is on Fo rather than Qx, because the error using the December 2008 SPE Journal

Fig. 16—One realization of channelized permeability distribution (left) and histogram (right). Log scale for permeability.

primitive model with ALG T* upscaling is relatively small. Shown in Fig. 18b are the results using EFBCs with ␭* and f* computed numerically for all coarse blocks. The error apparent in Fig. 18a is largely eliminated, demonstrating the applicability of EFBCs for this model. For this example, we investigate the effect of the size of the calibration data set. Figs. 18c and 18d present results applying the statistical estimation procedure using 10 realizations and one realization, respectively, for the calibration. In this work, realizations are selected randomly. For the one-realization calibration case, only one cluster was used to avoid generating clusters with very few points. The results in Figs. 18c and 18d again demonstrate only a slight degradation in accuracy relative to the case in which all ␭* and f* are computed numerically (Fig. 18b). In addition, the use of one realization for the calibration provides results that are nearly as accurate as those using 10 realizations. This is a significant observation as it indicates that a reasonable level of accuracy can be achieved for this case by computing pseudofunctions for only 1% of the coarse blocks in the 100 realizations. Our final set of results illustrates a more flexible application of the statistical estimation procedure. We introduce a thresholding technique to determine the coarse interfaces for numerical simulation or statistical estimation of ␭* and f*. The thresholding is based on single-phase coarse-scale flow rate. This treatment ensures that the ␭* and f* for coarse blocks in high-flow regions will be numerically simulated rather than estimated. Here, for each realization, we simulate ␭* and f* for coarse-block interfaces with flow rates that are greater than 80% of the maximum rate for that realization. In addition, we simulate all of the interfaces in one or more realizations to form the clusters used to predict the remaining upscaled functions. In the results presented here, we use three realizations in one case and one realization in the other case for this calibration. The results are shown in Fig. 19. In Fig. 19a, ␭* and f* are numerically computed for a total of 5.1% of the coarse interfaces, which includes simulations for both the calibration and for highflow interfaces; for Fig. 19b, ␭* and f* are numerically computed for 3.2% of the coarse interfaces. Note that without simulating the high flow interfaces, the numerically simulated ␭* and f* are 3% and 1%, respectively. We see that the results in Figs. 19a and 19b

Fig. 17—Oil cut for 100 realizations for channelized model and M=5. Black curves represent P50 and P10–P90 interval. December 2008 SPE Journal

are comparable to or slightly better than those in Figs. 18c and 18d, demonstrating the flexibility of the method. The impact and potential use of this thresholding procedure have been explored only briefly and warrant further investigation with other models. Discussion The results demonstrate that statistical procedures can be used to assign upscaled two-phase flow functions in coarse-scale models. The basic approach is applicable for any two-phase upscaling procedure and is very flexible in terms of the specific attributes used in the statistical estimation. The approach does, however, require that some amount of preprocessing be performed to determine the appropriate upscaling method and attributes for the problem under consideration. We now briefly quantify the computational speedups offered by the statistical estimation procedure. The speedups discussed here are relative to the full two-phase fine-scale simulation of all realizations. If global upscaling of the two-phase flow functions is applied, then the speedup will be caused solely by the savings achieved by avoiding the global simulation of some fraction of the fine-scale models. For example, if we simulate 10 of the 100 realizations to define the clusters and CDFs for ␭* and f* and use statistical estimation for the other 90 realizations, the speedup will be approximately a factor of 10 (as all other computations are very fast compared to the fine-scale simulations). More substantial speedups are achievable if we use local or extended local calculations for ␭* and f*. In our previous work

Fig. 18—Comparison of P50 (thick curves) and P10–P90 interval (thin curves) for oil cut between fine-scale (solid curves) and coarse-scale (dot-dash curves) models for channelized system and M=5. 409

Fig. 19—Comparison of P50 (thick curves) and P10–P90 interval (thin curves) for oil cut between fine-scale (solid curves) and coarse-scale (dot-dash curves) models for channelized system and M=5. The upscaled two-phase functions are numerically simulated in high-flow regions for all the realizations.

(Chen and Durlofsky 2006b), in which we numerically computed all of the coarse-block two-phase functions using extended local EFBC computations, overall speedups of 4 to 10 (average speedup factor of approximately 7) were observed. The bulk of the computational effort for the coarse-scale modeling was consumed by the two-phase upscaling computations. For present purposes, we assume a speedup factor of 7 for the case when all upscaled functions are computed numerically using EFBCs. Then, the speedup obtained when only 10% of these functions are computed numerically, with the rest estimated statistically, will be approximately a factor of 70. If only 3% of the upscaled two-phase flow functions are computed numerically (as in Fig. 19b), then speedups will be considerably greater (more than a factor of 200). These speedups will, of course, be affected by the details of the procedures, but the numbers cited here provide a general idea of the computational savings attainable using statistical estimation techniques. In this study, we considered a single set of relative permeabilities. If the fine-scale model contains facies characterized by different relative permeability curves or endpoints, then parameters associated with the various facies (e.g., volume fraction of particular facies, volume fraction-weighted endpoints) could be used as additional attributes. In this way, the fine-scale facies distribution would affect the upscaled two-phase flow functions used for the coarse model. Similarly, if different geological scenarios are to be considered, as opposed to multiple realizations of a particular scenario as considered here, parameters associated with the scenarios could also be used as attributes. In this work, we used a simple set of attributes consisting of the average and standard deviation of the fine-scale velocity, both computed during the single-phase parameter upscaling. As discussed earlier, it is possible to use other attributes to improve the clustering results. We note finally that deterministic (e.g., regression-type) procedures could also be applied to assign coarseblock functions. Conclusions The following conclusions can be drawn from this study: • A new approach, ensemble-level upscaling, was presented for efficiently generating upscaled two-phase flow functions for multiple geological realizations. In this approach, the upscaled two-phase functions are numerically calculated for only a small portion of coarse-scale blocks. For the great majority of coarse blocks in the multiple models, a statistical estimation procedure was introduced to estimate these functions on the basis of a cluster analysis of coarse-block attributes. These attributes used single-phase fine-scale velocity information, which is computed during the calculation of upscaled single-phase flow parameters. The procedure can be combined with any flow-based two-phase upscaling technique to significantly reduce computational costs. • The ensemble-level upscaling technique aims to achieve agreement at the ensemble level between the fine- and coarse-scale flow simulation models, rather than realization-by-realization agreement, as is the intent of most existing upscaling techniques. 410

For this purpose, the statistical estimation procedure was designed to maintain appropriate statistics (e.g., CDFs) of the upscaled two-phase flow functions. This provides an unbiased estimation of these functions and leads to unbiased predictions for the ensemble statistics of the fine-scale models. A high level of accuracy for individual realizations is not necessarily obtained. • The proposed method was applied to 2D synthetic models with different styles of heterogeneity and fluid-mobility ratios. It was shown that the method consistently corrected the biased predictions in the primitive coarse-scale models and was able to capture the ensemble statistics (e.g., P50 and P10–P90 interval) of the fine-scale models. The accuracy of the approach was found to be similar to that achieved when all of the upscaled two-phase flow functions were computed numerically (i.e., the underlying full flow-based method), but the computational cost was much less. Nomenclature f ⳱ Buckley-Leverett fractional flow function Fo ⳱ fractional flow of oil (oil cut) i ⳱ block index k ⳱ absolute permeability kb ⳱ average background permeability krj ⳱ relative permeability to phase j l ⳱ dimensionless correlation length L ⳱ size of local or global domain M ⳱ endpoint mobility ratio N ⳱ number of discretized saturation values p ⳱ pressure Q ⳱ total flow rate S ⳱ water saturation Sor ⳱ residual oil saturation Swc ⳱ connate water saturation t ⳱ time T ⳱ transmissibility u ⳱ Darcy velocity x ⳱ physical space coordinate ␦f ⳱ parameter to quantify upscaled fractional flow function ␦␭ ⳱ parameter to quantify upscaled total mobility ␭ ⳱ mobility ␮ ⳱ viscosity ␴ ⳱ standard deviation ␾ ⳱ porosity Subscripts j o w x, y

⳱ ⳱ ⳱ ⳱

phase oil (displaced fluid) water (displacing fluid) coordinate directions

Superscripts c ⳱ coarse scale * ⳱ upscaled or equivalent References Artus, V., Durlofsky, L.J., Onwunalu, J., and Aziz, K. 2006. Optimization of nonconventional wells under uncertainty using statistical proxies. Computational Geosciences 10 (4): 389–404. DOI:10.1007/s10596006-9031-9. Barker, J.W. and Dupouy, P. 1999. An analysis of dynamic pseudo-relative permeability methods for oil-water flows. Petroleum Geoscience 5 (4): 385–394. Barker, J.W. and Thibeau, S. 1997. A Critical Review of the Use of Pseudorelative Permeabilities for Upscaling. SPERE 12 (2): 138–143. SPE-35491-PA. DOI: 10.2118/35491-PA. Bishop, C.M. 1995. Neural Networks for Pattern Recognition. Oxford, UK: Oxford University Press. December 2008 SPE Journal

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Yuguang Chen is a lead research scientist on the Reservoir Simulation Research Team at Chevron Energy Technology Company in San Ramon, California. She holds MS and PhD degrees in petroleum engineering from Stanford University and a BS degree in engineering mechanics and an MS degree in fluid mechanics from Tsinghua University in Beijing, China. Chen was the winner of the SPE International Student Paper Contest (PhD level) in 2004. Louis J. Durlofsky is professor and chairman in the Department of Energy Resources Engineering at Stanford University. He was also affiliated with Chevron Energy Technology Company until January 2005. Durlofsky holds a BS degree from Penn State, as well as MS and PhD degrees from the Massachusetts Institute of Technology, all in chemical engineering. He received the SPE Reservoir Engineering Award in 2002 and the Lester C. Uren Award in 2007.

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